Transreal Arithmetic as a Consistent Basis For Paraconsistent Logics

Paraconsistent logics are non-classical logics which allow non-trivial and consistent reasoning about inconsistent axioms. They have been proposed as a formal basis for handling inconsistent data, as commonly arise in human enterprises, and as methods for fuzzy reasoning, with applications in Artificial Intelligence and the control of complex systems.

Formalisations of paraconsistent logics usually require heroic mathematical efforts to provide a consistent axiomatisation of an inconsistent system. Here we use transreal arithmetic, which is known to be consistent, to arithmetise a paraconsistent logic. This is theoretically simple and should lead to efficient computer implementations.

We introduce the metalogical principle of monotonicity which is a very simple way of making logics paraconsistent.

Our logic has dialetheaic truth values which are both False and True. It allows contradictory propositions, allows variable contradictions, but blocks literal contradictions. Thus literal reasoning, in this logic, forms an on-the-fly, syntactic partition of the propositions into internally consistent sets. We show how the set of all paraconsistent, possible worlds can be represented in a transreal space. During the development of our logic we discuss how other paraconsistent logics could be arithmetised in transreal arithmetic.